Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

293 questions with no upvoted or accepted answers
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7
votes
0answers
126 views

Find a flag to transform a matrix to an upper triangular one

Consider $F: \mathbb{R^3} \to \mathbb{R^3}$ represented by: $ A= \begin{bmatrix} 1 & 1 & 2 \\ -2 & 5 & 6 \\ 1 & -2 & -2 \\ \end{bmatrix} $ , eigenvalues: $...
6
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2answers
110 views

Can basis of kernel be extended to a Jordan basis?

Let $A\in\mathbb C^{n\times n}$ be nilpotent. A Jordan basis of $A$ is a basis of $\mathbb C^n$ with respect to which $A$ has Jordan normal form. Assume that we do not know the Jordan structure of $A$....
6
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0answers
1k views

An inverse of Jordan matrix - basis

Let $A\in M_{n\times n}$ be and invertible matrix over complex field and we assume it's already at Jordan form where $B=\{v_1,…,v_n \}$ is Jordan basis for A. Find Jordan form and Jordan basis for $...
5
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2answers
57 views

Have I found the Jordan form correctly?

I am given that the minimal polynomial and characteristic polynomial of a matrix are both $(x-1)^2(x+1)^2$. I have found the Jordan form to be $$\begin{bmatrix}1&1&0&0\\0&1&0&0\...
5
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0answers
250 views

Lost on rational and Jordan forms

I'm having a lot of trouble trying to understand rational canonical form, primary rational canonical form, and Jordan form. I've looked at the posts about this, but I haven't been able to understand ...
5
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2answers
897 views

Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in \...
5
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2answers
2k views

Jordan form exercise

What am I doing wrong? I've been learning how to put matrices into Jordan canonical form and it was going fine until I encountered this $4 \times 4$ matrix: $A=\begin{bmatrix} 2 & 2 & 0 &...
4
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0answers
451 views

Jordan Block of Kronecker Product

Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\...
4
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1answer
293 views

Why block-diagonal form for nilpotent matrices?

I am currently reading Jim Hefferon's Linear Algebra. In chapter 5, nilpotence, strings, he goes through the process of finding a string basis of a map, and proves that there exists a string basis ...
4
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0answers
2k views

Prove the direct sum of generalized eigenspaces is the whole vector space

Given a $n\times n$ matrix $A$ over an algebraically closed field, let $\lambda_1,...,\lambda_k$ be its eigenvalues, and let $V_{\lambda_i}$ be the generalized eigenspace of $\lambda_i$. The question ...
3
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33 views

What is the fastest way to find the characteristic polynomial of $4\times4$ matrix and its change of basis matrix?

Let: $$A=\begin{pmatrix} 3 & 0 & -2 & -3\\ 4 & -8 & 14 & -15\\ 2 & -4 & 7 & -7\\ 0 & 2 & -4 & 3 \end{pmatrix}$$ Find a change of basis matrix $P$ such ...
3
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35 views

Jordan form with -1 in block

I'm having troubles finding the right base, I keep getting a weird Jordan form. I need to find the Jordan form and basis of $$ \left[\begin{array}{cc} 1 & 1\\ -1 & 3 \end{array}\right] $$ this ...
3
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0answers
60 views

Upper bound of the remainder term of $\int_0^t e^{A \tau} d\tau$ using Lagrange Remainder of the Taylor series

$\newcommand{\mat}[1]{\begin{bmatrix}#1\end{bmatrix}}$ $\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$Let $A$ be an $n\times n$ real matrix which has real block diagonal form and each block has a ...
3
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1answer
50 views

Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero?

Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero? if there is a proper number 0, then you can try to find a matrix in the form of J^(-...
3
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2answers
69 views

Difficulties with Jordan normal form

i'm studying in German and because of corona virus we had only a video lecture, so unfortunately I have not understood how to deal in cases when I do not have the matrix. If I had it, I think I got ...
3
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1answer
110 views

Possible Jordan Canonical Forms

Suppose I have a matrix $A \in M_{n \times n}(\mathbb{C})$ such that its minimal polynomial is either $x-1$ or $(x-1)^{2}$. What are its possible Jordan Canonical Forms? I was thinking that if its ...
3
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0answers
56 views

minimal polynomial of a matrix B given minimal polynomial of $B^2$

If we are given a minimal polynomial for a matrix $B^2$ can we deduce the minimal polynomial for $B$ $?$ Example: if the minimal polynomial for $B^2$ is $m(\lambda) = \lambda^4$ then can we deduce ...
3
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60 views

Find the Jordan Canonical Form of the given transformation

The transformation here is $T(f(x)) = f(x + 1) + f(x − 1)$ which is a linear endomorphism on V, where $V={f(x) ∈ R[x] : deg f(x) ≤ 2017}$ So I have to find the jcf J of T. Along with a basis of B. $...
3
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1answer
412 views

Minimal polynomial and possible Jordan forms

Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan ...
3
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211 views

Classification of bilinear forms: operator $A^{-1} A^T$ for bilinear form $A$

I would like to understand a classification of non-degenerate (not necessary symmetric or skew-symmetric) bilinear forms over an algebraically closed field via an operator $\kappa=A^{-1} A^T$ for a ...
3
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0answers
2k views

Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
3
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0answers
395 views

Jordan normal form theorem proof question

Theorem: Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan blocks....
3
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0answers
41 views

Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which $\mathcal{M}_{\mathcal{S}...
3
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1answer
700 views

relation between minimal polynomial and jordan normal form

I just solved some exercises on minimal polynomials and i remember that there is a relation between the minimal polynomial and the jordan normal form. But my question is the following : knowing the ...
3
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0answers
1k views

Jordan Canonical Form and Minimal Polynomial

I was wondering what the relationship between the minimal polynomial and the Jordan Canonical Form is. Given a matrix, all one needs to do is to compute the characteristic polynomial to determine the ...
2
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0answers
32 views

Switch from 1 to random real in Jordan Decomposition

Context : Let's suppose $L$ is a linear map from $\mathbb{R^k}\rightarrow \mathbb{R}^k$ , $k$ strictly positive integer. Let's suppose $\epsilon$ is a strictly positive real. In an exercice , i have ...
2
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1answer
65 views

How to determine Jordan form, by knowing dimension of Kernel

I have the following information: $p_A(\lambda) = (\lambda - 1)^6(\lambda + 2)^4$ and $m_A(\lambda) = (\lambda - 1)^3(\lambda + 2)^2$ and also $\dim(\operatorname{Ker}(A-I)^2) = 5$ and $\dim(\...
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0answers
46 views

Finding generalized eigenvectors of a matrix

I would like to know how to find the generalized eigenvectors to the following matrix $A$, so that I can express $A$ as $PJP^{-1}$. $$ A = \begin{bmatrix} 1 & -3 & 1\\ 1 & 5 & -1\\2 &...
2
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0answers
31 views

Confusion about the Jordan-Chevalley/Dunford decomposition in $\mathbb{R}$, example of a rotation (solved!!!)

I'm writing some notes on Jordan-Chevalley decompositions in which I want to treat both the real and complex case in one statement. One could of course write as in the french wikipedia article: an ...
2
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0answers
59 views

Connection between uniqueness of Jordan normal form and Jordan–Chevalley decomposition

Let $f$ be an endomorphism of a finite-dimensional vector space $V$. Definition. A Jordan–Chevalley decomposition of $f$ is a decompositon $f = d + n$ where $d$ and $n$ are endomorphisms of $V$ such ...
2
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0answers
56 views

Looking for an intuition of the definition of Generalized Eigenspaces

The eigenspace of (a square matrix) $A$ corresponding to $\lambda$ is the collection of all vectors $\mathbf{x}$ that satisfy $A\mathbf{x}=\lambda\mathbf{x}$, or equivalently, $(A-\lambda I)\mathbf{x}=...
2
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52 views

Jordan normal form in a reductive group

Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
2
votes
1answer
30 views

find all the matrices (which are not similiar) which fulfill this formula

I need to find all the matrices $A\in M_{4x4}\left(\mathbb{C}\right)\:$ such that: $$A^4-2A^2+I\:=\:0$$ which means $\left(A^2-I\right)^2=0$ So I see that there is a few groups of which can give ...
2
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0answers
79 views

Bounding 2-norm of powers of a matrix

Suppose that $A$ is a $n \times n$ matrix with $\rho(A) \leq 1$ and $\|A\|_2 \leq R$, where $R>1$. How can I show an upper bound on $\|A^k\|_2$ that is polynomial in $k$? A trivial upper bound is ...
2
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0answers
88 views

Jordan normal form of upper bidiagonal matrix

I am trying to find the Jordan normal form of a general matrix form, which is an $N \times N$ real upper bidiagonal matrix with non-zero diagonal entries and rows that sum to one: $$\mathbf{M} = \...
2
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0answers
48 views

Properties of blocks in blockmatrix A if the matrix pair (E,A) is regular

I am working on a problem from the book "Differential-Algebraic Equations: Analysis and Numerical Solution" by Kunkel, Mehrmann. It is the Exercise 3 from Page 53 concerning matrix pairs $(E,A)$ and ...
2
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1answer
17 views

Basics of Jordan matrix, please clarify the following

Let $$J=\oplus_{i=1}^{k} J_{n_i}(\lambda)$$ where $J_{n_i}(\lambda)$ is a jordan block of size $n_i$ with $\lambda $ on its diagonal, and $\sum_{i=1}^{k}n_i = n $, so $J$ is $n\times n$ matrix, ...
2
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0answers
36 views

How many similarity classes are zeroes of a given polynomial.

I am looking for an easy way of calculating the number of similarity classes of complex matrices that satisfy some polynomial $p(t)$. As an example consider $p(t)=(t-1)^3(t+1)^4$ and $5\times 5$ ...
2
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0answers
50 views

Analogy of Jordon Normal Form for Antilinear Maps

Given complex vector spaces $V$, and antilinear $T:V \rightarrow V$, then if we fix a basis of $V$, we can represent $T$ by the matrix of the linear $T \circ J$, where $J$ is complex conjugation. I ...
2
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0answers
67 views

How can I find the Jordan form of this upper triangular Toeplitz matrix?

Given an $n \times n$ matrix $A$ whose $(i,j)$ entry is $$a_{ij} = \begin{cases} n-j+i & \text{if } j \geq i\\ 0 & \text{otherwise}\end{cases}$$ find its Jordan form. I know that all the ...
2
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0answers
32 views

For which values ​the matrix is ​diagonalizable

For which values ​​of $a$ matrix $A$ is ​​diagonalizable? $$A = \pmatrix{0&i\\i&a}$$ in the case that it is not diagonalizable determine a base of Jordan Attempt: The minimal polynomial ...
2
votes
1answer
277 views

Linear Algebra : Jordan Canonical form (Jordan blocks and the Super-Diagonal)

In terms of Jordan Canonical Form, and more specifically about Jordan Blocks. When there is a definition about Jordan Blocks they say the eigenvalues go on the principle diagonal and the diagonal ...
2
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0answers
109 views

When is possible to use an orthogonal matrix to put in Jordan form a matrix?

I know that if I have a symmetrical matrix defined on $R$, it is always diagonalisable and I can always find beetwen the matrix of its eigenspaces an orthogonal matrix. While if I have a non ...
2
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0answers
149 views

Uniqueness of Jordan-Chevalley-Decomposition - why do the nilpotent matrices commute?

The Jordan-Chevalley-Theorem states that for a given endomorphism $f\in \mathrm{End}_K(V)$ of a $K$-vector space $V$, such that the characteristic polynomial $\chi_f$ splits into factors, there exist ...
2
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0answers
146 views

Square root of a matrix by looking at the eigenspaces

A problem that is well known in linear algebra is the existence of square root of a matrix, where square root of matrix $A \in M_n(\mathbb F)$ is defined to be $K \in M_n(\mathbb F)$ such that $K^2=A$ ...
2
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0answers
74 views

Trouble in proof of Jordan Canonical Theorem

Jordan Canonical Theorem stated that: Let $K$ be an algebra closed field. Let $V$ be a nonzero, finite dimensional vector space over $K$, and let $\psi \in \operatorname{End}_K(V)$. Then there ...
2
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0answers
78 views

Classification of Matrices and normal forms

As is shown in couses of Linear Algebra, for every square matrix $A$ one can choose $S,T,P\in GL(n,K)$ so that $SAT^{-1}=\operatorname{diag}(1,...1,0,...,0)$ and $PAP^{-1}$ is in Jordan normal form. ...
2
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0answers
63 views

Polar Decomposition in Real Algebraic Groups

Every element $g \in GL(n,\mathbb{C})$ has a unique Jordan decomposition $$ g = g_u g_s $$ where $g_u$ is unipotent, $g_s$ is semisimple (i.e. diagonalizable over $\mathbb{C})$ and $g_ug_s=g_sg_u$. It ...
2
votes
0answers
73 views

Calculate $\det(p(A))$

Let $n \in \mathbb N$, $A \in \mathbb C^{n \times n}$ be nilpotent and $l\in \mathbb N$. Further, let $$p = \sum_{i=0}^n \alpha_i A^i \in \mathbb C[t]$$ be a polynomial. Show that zero is the only ...
2
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0answers
161 views

Problem on characteristic polynomial and minimal polynomial

I am given two matrices a and b such that characteristic polynomial and minimal polynomial of a and b are equal I have to check If they are similar JC form of a and b are same Rank(a) = rank(b) If ...

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